Group classification of (1+1)-Dimensional Schrödinger Equations with Potentials and Power Nonlinearities

Popovych, Roman O.; Ivanova, Nataliya M.; Eshraghi, Homayoon

Group classification of (1+1)-Dimensional Schrödinger Equations with Potentials and Power Nonlinearities

Authors

Popovych, Roman O.

Ivanova, Nataliya M.

Eshraghi, Homayoon

Description

We perform the complete group classification in the class of nonlinear Schrödinger equations of the form $iψ_t+ψ_{xx}+|ψ|^γψ+V(t,x)ψ=0$ where $V$ is an arbitrary complex-valued potential depending on $t$ and $x,$ $γ$ is a real non-zero constant. We construct all the possible inequivalent potentials for which these equations have non-trivial Lie symmetries using a combination of algebraic and compatibility methods. The proposed approach can be applied to solving group classification problems for a number of important classes of differential equations arising in mathematical physics.
10 pages